 Solution

The variable $\psi$ is a rotating-flow pressure potential including the conservative body-force and centrifugal potentials. For constant mass density,up to an additive function of time, where $\Phi$ is any imposed conservative potential. The last term is the centrifugal potential; its negative gradient is the centrifugal acceleration. The Coriolis force remains explicit in the equation because it is velocity-dependent and cannot be absorbed into a scalar position potential.

The further constraint is incompressible flow:For the specified shear, ${\mathbf{u}}_0\cdot \nabla {\mathbf{u}}_0=0$ and ${\nabla }^2{\mathbf{u}}_0=0$. Its Coriolis acceleration is $4\Omega Ax{\mathbf{e}}_x$, so the corresponding total basic pressure potential can be chosen as ${\psi }_0=-2\Omega A{x}^2$. This supplies the force balance required for the basic state used below.